0708-1300/Class notes for Tuesday, October 9

Class Notes

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

Scanned Notes

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Typed Notes - First Hour

Reminder:

1) An immersion locally looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n\rightarrow\mathbb{R}^m} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\mapsto(x,0)}

2) A submersion locally looks like $\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}$ given by $(x,y)\mapsto x$

Today's Goals

1) More about "locally things look like their differential"

2) The trick Sard's Theorem: "Evil points are rare, good points everywhere"

Definition 1

Let $f:M^{m}\rightarrow N^{n}$ be smooth. A point $p\in M^{m}$ is critical if $df_{p}$ is not onto $\Leftrightarrow$ rank $df_{p}<n$. Otherwise, p is regular.

Definition 2 A point $y\in N^{n}$ is a critical value of f if $\exists p\in M^{m}$ such that p is critical and $f(p)=y$. Otherwise, y is a regular value

Example 1

Consider the map $f:S^{2}\rightarrow \mathbb {R} ^{2}$ given by $(x,y,z)\mapsto (x,y)$. I.e., the projection map. The regular points are all the points on $S^{2}$ except the equator. The regular values, however, are all $(x,y)$ such that $x^{2}+y^{2}\neq 1$

Example 2

Consider $f:\mathbb {R} ^{3}\rightarrow \mathbb {R}$ given by $p\mapsto ||p||^{2}$. That is, $(x,y,z)\mapsto x^{2}+y^{2}+z^{2}$. Clearly $df|_{p=(x,y,z)}=(2x,2y,2z)$ and so p is regular $\Leftrightarrow df_{p}\neq 0\Leftrightarrow p\neq 0$

So, the critical values are the image of zero, thus only zero. All other Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in\mathbb{R}} are regular values.

Note: In both the last two examples there were points in the target space that were NOT hit by the function and thus are vacuously regular. In the previous example these are the point x<0.

Example 3

Consider a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} from a segment in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} onto a curve in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\gamma} is never zero. Thus, rank(Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\gamma = 1} ) and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\gamma} is never onto. Hence, ALL points are critical in the segment. The points on the curve are critical values, as they are images of critical points, and all points in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} NOT on the curve are vacuously regular.

Theorem 1

Sard's Theorem

Almost every $y\in N^{n}$ is regular Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow} the set of critical values of f is of measure zero.

Note: The measure is not specified (indeed, for a topological space there is no canonical measure defined). However the statement will be true for any measure.

Theorem 2

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M^m\rightarrow N^n} is smooth and y is a regular value then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} is an embedded submanifold of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M^m} of dimension m-n.

Re: Example 2

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} is a sphere and hence (again!) the sphere is a manifold

Re: Example 3

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(y)} for regular y is empty and hence we get the trivial result that the empty set is a manifold

Proof of Theorem 2

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:M^m\rightarrow N^n} is smooth and y is a regular value. Pick a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in f^{-1}(y)} . p is a regular point and thus $df_{p}$ is onto. Hence, by the submersion property (Reminder 2) we can find a "good charts" thats maps a neighborhood U of p by projection to a neighborhood V about y. Indeed, on U f looks like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n\times\mathbb{R}^{m-1}\rightarrow \mathbb{R}^n} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,z)\mapsto x} .

SoFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}(0) = \{(0,z)\} = \mathbb{R}^{m-n}} . Q.E.D

Diversion

Arbitrary objects can be described in two ways:

1) With a constructive definition

2) with an implicit definition

For example, a constructive definition of lines in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{v_1 + tv_2\}} but implicitly they are the solutions to the equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax+by+cz = d} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ez+fy+gz+h} .

Hence in general, a constructive definition can be given in terms of an image and an implicit definition can be given in terms of a kernal.

Homological algebra is concerned with the difference between these philosophical approaches.

Remark

For submanifolds of smooth manifolds, there is no difference between the methods of definition.

Definition 3

Loosely we have the idea that a concave and convex curve which just touch at a tangent point is a "bad" intersection as it is unstable under small perturbation where as the intersection point in an X (thought of as being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} ) is a "good" intersection as it IS stable under small perturbations.

Precisely,

Let $N_{1}^{n_{1}},N_{2}^{n_{2}}\subset M$ be smooth submanifolds. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in N_1^{n_1} \bigcap N_2^{n_2}}

We say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_1} is transverse to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_2} in M at p if for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_1\subset T_p M} andFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_2 \subset T_p M} satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p N_1 + T_p N_2 = T_p M}

Example 4

Our concave intersecting with convex curve example intersecting tangentially has both of their tangent spaces at the intersection point being the same line and thus does not intersect transversally as the sum of the tangent spaces is not all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2} .

Our X example does however work.

Typed Notes - Second Hour

Coming soon to a wikipedia near you.